Conway's Game of Life is a two-dimensional cellular automaton. As a dynamical system, it is well-known to be computationally universal, i.e.\ capable of simulating an arbitrary Turing machine. We show that in a sense taking a single backwards step of Game of Life is a computationally universal process, by constructing patterns whose preimage computation encodes an arbitrary circuit-satisfaction problem, or (equivalently) any tiling problem. As a corollary, we obtain for example that the set of orphans is coNP-complete, exhibit a $6210 \times 37800$-periodic configuration that admits a preimage but no periodic one, and prove that the existence of a preimage for a periodic point is undecidable. Our constructions were obtained by a combination of computer searches and manual design.

翻译：康威的生命游戏是一种二维元胞自动机。作为一个动力学系统，它已知具有计算普适性，即能够模拟任意图灵机。我们通过构造这样的模式——其原像计算编码了任意电路满足问题（或等价地，任意平铺问题）——表明，在某种意义上，生命游戏的单步逆向过程是一个计算普适的过程。作为推论，我们得到例如“孤儿”集合是coNP完全的，展示了一个周期为$6210 \times 37800$的构型，它有一个原像但不具有周期性的原像，并证明了周期点的原像存在性是不可判定的。我们的构造是通过计算机搜索与人工设计相结合的方式获得的。